A slew of sangaku problems deal with chains of inscribed circles, see, for example, Steiner's sangaku. The elegant sangaku below is the simplest in the collection by Fukagawa and Pedoe (1.8.6), yet it gives a chance to discuss a formula that was not used so far anywhere else at the site.

Solution

Copyright © 1996-2008 Alexander Bogomolny

This sangaku dates from 1814 and was written in the Gumma prefecture.

The derivation of a more traditional formula for the common chain of circles inscribed in an arbelos

rn = ts / (n² + t + t²).

is easily adapted to the present case. The result is the following formula

(*)	rn = 4ts / ((2n - 1)² + 4t + 4t²),

where t = r / (s - r). Denoting S = 4t + 4t² we can write

r1 = 4s / (1 + S), r4 = 4s / (7² + S), r7 = 4s / (13² + S),

which we plug into the required identity

7 / r4 = 2 / r7 + 5 / r1

getting

7(7² + S) = 2(13² + S) + 5(1 + S)

which simplifies to

7·7² = 2·13² + 5.

As one can easily verify, both sides are equal to 343.

Sangaku

1. Sangaku: Reflections on the Phenomenon
2. Critique of My View and a Response
3. 7
4. 3-4-5 Triangle by a Kid
5. 7
6. A 49^th Degree Challenge
7. A Geometric Mean Sangaku
8. A Hard but Important Sangaku
9. A Sangaku: Two Unrelated Circles
10. A Sangaku by a Teen
11. A Sangaku Follow-Up on an Archimedes' Lemma
12. A Sangaku with an Egyptian Attachment
13. A Sangaku with Many Circles and Some
14. An Old Japanese Theorem
15. Archimedes Twins in the Edo Period
16. Arithmetic Mean Sangaku
17. Bottema Shatters Japan's Seclusion
18. Circles and Semicircles in Rectangle
19. Circles in a Circular Segment
20. Circles Lined on the Legs of a Right Triangle
21. Equal Incircles Theorem
22. Equilateral Triangle, Straight Line and Tangent Circles
23. Equilateral Triangles and Incircles in a Square
24. Five Incircles in a Square
25. Four Hinged Squares
26. Four Incircles in Equilateral Triangle
27. Gion Shrine Problem
28. Harmonic Mean Sangaku
29. Heron's Problem
30. In the Wasan Spirit
31. Incenters in Cyclic Quadrilateral
32. Japanese Art and Mathematics
33. Malfatti's Problem
34. Maximal Properties of the Pythagorean Relation
35. Neuberg Sangaku
36. Out of Pentagon Sangaku
37. Peacock Tail Sangaku
38. Pentagon Proportions Sangaku
39. Pythagoras and Vecten Break Japan's Isolation
40. Radius of a Circle by Paper Folding
41. Review of Sacred Mathematics
42. Sangaku ŕ la V. Thebault
43. Sangaku and The Egyptian Triangle
44. Sangaku in a Square
45. Sangaku Iterations, Is it Wasan?
46. Sangaku with 8 Circles
47. Sangaku with Three Mixtilinear Circles
48. Sangaku with Versines
49. Sangakus with a Mixtilinear Circle
50. Sequences of Touching Circles
51. Square and Circle in a Gothic Cupola
52. Tangent Circles and an Isosceles Triangle
53. The Squinting Eyes Theorem
54. Steiner's Sangaku
55. Three Incircles In a Right Triangle
56. Three Squares and Two Ellipses
57. Three Tangent Circles Sangaku
58. Triangles, Squares and Areas from Temple Geometry
59. Two Arbelos, Two Chains
60. Two Circles in an Angle

• Arbelos - the Shoemaker's Knife
• 7
• Another Pair of Twins in Arbelos
• Archimedes' Quadruplets
• Archimedes' Twin Circles and a Brother
• Book of Lemmas: Proposition 5
• Book of Lemmas: Proposition 6
• Chain of Inscribed Circles
• Concurrency in Arbelos
• Concyclic Points in Arbelos
• Ellipse in Arbelos
• Gothic Arc
• Pappus Sangaku
• Rectangle in Arbelos
• Squares in Arbelos
• The Area of Arbelos
• Twin Segments in Arbelos
• Two Arbelos, Two Chains

Copyright © 1996-2008 Alexander Bogomolny